The England Under-21 international moved to White Hart Lane from Sporting Lisbon over the summer and made an instant impact as he grabbed the only goal of the game against their local rivals.
Dier appeared on the edge of the West Ham box in added time to secure three points for the visitors after both sides had been reduced to 10 men.
Kyle Naughton was dismissed for handball, with Mark Noble spurning the chance to score from the resulting penalty, before the Hammers had James Collins sent off in the second half.
Dier was delighted with his start at Spurs and was happy to be involved on the opening day of the new Barclays Premier League season.
"I was just really happy to play to be honest and to score on top of that, well, I never imagined it," he said.
"I saw the keeper coming out in the corner of my eye and I wanted to touch the ball around him and put it in
It all happened really quickly
I didn't get any time to think about it and that was probably a good thing.
"I really enjoyed it and it's why I wanted to come here, to play in games like this
I'm really happy to be here."
Many would have been surprised to see Dier appear on the edge of the West Ham box in stoppage time but Tottenham boss Mauricio Pochettino believes it shows how his side want to play this season.
"This is a picture of what is our philoshopy, our style," he told Tottenham's official website.
"It is important as it shows our ambition and our philosophy to try and win.
"He was brilliant, but not only him, the whole team
It was an unbelievable win because last season we came here and left with zero points."
West Ham handed two of their own summer signings their debuts as Cheikhou Kouyate and Aaron Cresswell were named in Sam Allardyce's starting line-up.
Enner Valencia also made his bow with a late substitute appearance but it was former Ipswich full-back Cresswell who particularly caught the eye.
The 24-year-old was named in the Professional Footballers' Association Championship Team of the Year last season but was left deflated following his impressive debut.
"I had all my family here and it was a proud moment for me personally," he said.
"But it's not about me, it's about the team and getting the three points
It's not personal, it's not about individual performances
The first thing you look at is the result and we didn't get that on Saturday, so I'm very disappointed."
The Hammers face another London derby when they travel to Crystal Palace next weekend and Cresswell is keen to get the season up and running.
"We'll sit down, go over it, see what we can take out of the game and where we could have gone on to win it," he added.
"It's just about taking the positives into the next game and looking forward to it."
Source : PA
Source: PA